Question

current attempt in progress
two identical satellites are in orbit about the earth. one orbit has a radius r and the other 2r. the centripetal force on the satellite in the larger orbit is ______ as that on the satellite in the smaller orbit.
earth
o half as great
o four times as great
o the same
o one fourth as great
o twice as great

Explanation:

Step1: Recall centripetal - force formula

The centripetal force $F = \frac{GMm}{r^{2}}$ for a satellite of mass $m$ orbiting the Earth of mass $M$ with orbital radius $r$ (where $G$ is the gravitational constant).

Step2: Calculate force for smaller - radius orbit

Let $F_1$ be the centripetal force for the orbit of radius $r$. So, $F_1=\frac{GMm}{r^{2}}$.

Step3: Calculate force for larger - radius orbit

Let $F_2$ be the centripetal force for the orbit of radius $2r$. Then $F_2=\frac{GMm}{(2r)^{2}}=\frac{GMm}{4r^{2}}$.

Step4: Find the ratio of forces

$\frac{F_2}{F_1}=\frac{\frac{GMm}{4r^{2}}}{\frac{GMm}{r^{2}}}=\frac{1}{4}$.

Answer:

D. one fourth as great